One of the first things you could do is put a hasp right on the front of this cabinet, but that doesn't exactly look too good. How do you take a nice piece of furniture and put a nice lock in it so it looks good, is functional but also very secure. You can use something like this. This is something like a desk lock. A basic key, a lock that goes through the door and a key is used to latch the door shut. We want to have this centered but you also have to have the "L" shape lock close properly.
This lock is a 3/4" lock, so we take a drill and start drilling the hole for it.
When you are half way through you need to drill the rest from the back side so you don't break the door.
Then put the lock through the door and make sure you keep your finger on the front so the pins don't fall out. Then you put a nut on the lock and tighten it with a wrench. Then put all the other pieces back on.
Now to keep the other door shut while this one is locked is pretty simple. Instead of adding a another lock, we just add this little piece of metal the other side. That way when this one is locked the other can't be opened.